Differentiate the following w.r.t. as indicated :
`x^(2)" w.r.t. "x^(3)`.

Answer

To differentiate \( x^2 \) with respect to \( x^3 \), we can use the chain rule. We will denote \( u = x^2 \) and \( v = x^3 \). We need to find \( \frac{du}{dv} \).

### Step-by-step Solution:

1. **Define \( u \) and \( v \)**:
\[ u = x^2 \quad \text{and} \quad v = x^3 \]

2. **Differentiate \( u \) with respect to \( x \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(x^2) = 2x \]

3. **Differentiate \( v \) with respect to \( x \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(x^3) = 3x^2 \]

4. **Use the chain rule to find \( \frac{du}{dv} \)**:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{2x}{3x^2} \]

5. **Simplify the expression**:
\[ \frac{du}{dv} = \frac{2}{3x} \]

### Final Answer:
\[ \frac{du}{dv} = \frac{2}{3x} \]